Chapter 9: Vector Calculus
Section 9.4: Differential Identities
Example 9.4.3
Show that for sufficiently well-behaved functions fx,y,z, the gradient of f is curl-free.
Solution
Mathematical Solution
The gradient of fx,y,z is the vector ∇f=fxfyfz and its curl is the vector
∇×∇f=|ijk∂x∂y∂zfxfyfz| = ∂y(fz)−∂z(fy)∂z(fx)−∂x(fz)∂x(fy)−∂y(fx) = fzy−fyzfxz−fzxfyx−fxy = 000
The curl vanishes because of the equality of the mixed partial derivatives, guaranteed, for example, by continuity of the second partial derivatives.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Tools≻Tasks≻Browse: Calculus - Vector≻ Vector Algebra and Settings≻ Display Format for Vectors
Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Obtain the curl of the gradient of f
Common Symbols palette: Del and cross-product operators
Context Panel: Evaluate and Display Inline
∇×∇fx,y,z =
Maple Solution - Coded
Load the Student VectorCalculus package and execute the BasisFormat command.
withStudent:-VectorCalculus:BasisFormatfalse:
Apply the Gradient and Curl commands from the Student VectorCalculus package.
CurlGradientfx,y,z =
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